Question

On the argand plane, let $\alpha=-2+3 z, \beta=-2-3 z \&|z|=1$. Then the correct statement is -

• A. $\alpha$ moves on the circle, centre at $(-2,0)$ and radius 3
• B. $\alpha \& \beta$ describe the same locus
• C. $\alpha \& \beta$ move on different circles
• D. $\alpha-\beta$ moves on a circle concentric with $|z|=1$

Answer 1

Answer: (A), (B), (D)

$\alpha=-2+3 z$
$\alpha+2=3 z$
$|\alpha+2|=3|z|$
$(x+2)^2+y^2=9$
Similarly $\quad \beta=-2-3 z$
$\Rightarrow \beta+2=-3 z $
$\Rightarrow |\beta+2|=|-3 z| $
$ (x+2)^2+y^2=9$
Now $\alpha-\beta=6 z \Rightarrow|\alpha-\beta|=6|z|$
so $(\alpha-\beta)$ moves on a circle with centre as origin and radius 6 .